Prime numbers Jaap Top

Prime numbers

Jaap Top

IWI-RuG & DIAMANT

12 March 2007

Prime

• Latin ‘primus’, Indo-European ‘per-’,‘pre-’

• first, most basic

• (primitive), indecomposable

• the fundamental building blocks for numbers

Every integer number is a product of primes.

101 = 101

1001 = 7 × 11 × 13 10001 = 73 × 137 100001 = 11 × 9091 1000001 = 101 × 9901 10000001 = 11 × 909091 100000001 = 17 × 5882353

1000000001 = 7 × 11 × 13 × 19 × 52579 10000000001 = 101 × 3541 × 27961

100000000001 = 11 × 11 × 23 × 4093 × 8779 1000000000001 = 73 × 137 × 99990001

10000000000001 = 11 × 859 × 1058313049

100000000000001 = 29 × 101 × 281 × 121499449

1000000000000001 = 7 × 11 × 13 × 211 × 241 × 2161 × 9091 etc.(?)

Euclid (±330BC–???), Elements, Book IX, Proposition 20:

There are infinitely many prime numbers

(. . . more than any given number . . .)

Proof (Euclid). More than any given number???

Suppose we have some number of primes

Multiply all of them, and after that, add 1 to the product The result is a new number (larger than the primes we had) If the new number is prime, we found a new prime!

If the new number is not a prime, then it is composite, hence some prime number divides it

This prime is not among the old ones, since they divide the product, hence leave a remainder 1 when you divide the product plus 1 by them

So also in this case, we found a new prime!

How to find all primes?

Not by Euclid’s proof !

Trick of Eratosthenes (276 BC–194 BC)

• Write the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, . . .

• The first one, 2, is prime. Cross out its multiples (they are not prime):

2, 3, 6 4, 5, 6 6, 7, 6 8, 9, 610, 11, 612, 13, 614, 15 . . .

• The next number (3) is prime (if it was a multiple of a smaller one, it was crossed out). Cross out its multiples

2, 3, 6 4, 5, 6 6, 7, 6 8, 6 9, 610, 11, 612, 13, 614, 615, . . .

• et cetera. What remains, are all primes

Euclid’s theorem says, that no matter how far you go, you will find numbers that are not crossed out

There exist arbitrarily long intervals of crossed-out integers

In other words, there exist consecutive primes whose difference is larger than any given number

Proof: fix a (large) number n and multiply:

M := 2 × 3 × 4 × 5 × . . . × n.

Now M + 2 is a multiple of 2, and M + 3 is a multiple of 3, and M + 4 is a multiple of 4, et cetera, until M + n, which is a multiple of n

So we constructed a large interval not containing any primes!

Example: 90, 91 = 7 × 13, 92, 93 = 3 × 31, 94, 95, 96.

How many primes below a given number?

below: number of primes:

100 25

10, 000 1, 229

1000, 000 78, 498

100, 000, 000 5, 761, 455

10, 000, 000, 000 455, 052, 511

1000, 000, 000, 000 37, 607, 912, 018 100, 000, 000, 000, 000 3, 204, 941, 750, 802 10, 000, 000, 000, 000, 000 279, 238, 341, 033, 925 1000, 000, 000, 000, 000, 000 24, 739, 954, 287, 740, 860 100, 000, 000, 000, 000, 000, 000 2, 220, 819, 602, 560, 918, 840 10, 000, 000, 000, 000, 000, 000, 000 201, 467, 286, 689, 315, 906, 290 What percentage of all numbers is prime? Table:

25, 12, 7.8, 5.8, 4.6, 3.8, 3.2, 2.8, 2.5, 2.2, 2.0, . . . (??)

Carl Friedrich Gauss (1777-1855)

Notation: π(n) means: the number of primes below n

So, for example, π(100) = 25 and π(12032007) = 790009 Gauss (age 15 . . .) ‘guessed’: π(n) is approximately n

ln(n)

Meaning: for larger and larger n, the quotient of the two expres- sions tends to 1

ln(n): ‘natural logarithm’ of n (≈ number of digits times 2.3)

π(x) and x/ln(x):

This guess by Gauss is called ‘the prime number theorem’

Proven in 1896 by Jacques Solomon Hadamard (1865–1963) and also by the Belgian mathematician Charles Jean Gustave Nicolas Baron de la Vall´ee Poussin (1866–1962)

Hadamard & De la Vall´ee Poussin

Some consequences:

• Approximately 100/ ln(n) percent of all numbers below n is prime

• 0 percent of all numbers is prime

• The probability of picking a prime from the set of numbers below n, is approximately 1/ln(n)

• Of all numbers of 10, 000, 000 digits, roughly 0.000004% is prime

The next EFF prize is expected to be awarded this year:

GIMPS = Great Internet Mersenne Prime Search

Current record: 2^32,582,657 − 1 is prime (9,808,358 digits)

Mersenne primes: primes of the form power of 2, minus 1

Euclid: such primes give ‘perfect numbers’ (perfect means: equal to the sum of their divisors [‘parts’] )

Examples: 2² − 1 = 3 is prime, 2 × 3 = 6 = 1 + 2 + 3 is perfect 2³ − 1 = 7 is prime, 2² × 7 = 28 = 1 + 2 + 4 + 7 + 14

2⁵ − 1 = 31 is prime, 2⁴ × 31 = 496 = 1 + 2 + . . . + 16 + 31 + 62 + . . . + 248

Marin Mersenne stated (1544) incorrectly: 2⁶⁷ − 1 and 2²⁵⁷ − 1 are primes. Also, incorrectly, that 2⁶¹ − 1 and 2⁸⁹ − 1 are not primes

2⁶⁷ − 1 = 193707721 × 761838257287

Marin Mersenne (1588–1648)

For testing whether M := 2^p− 1 is prime, there is a test involving less than p steps:

• s₁ := 4

• For n = 2, . . . , p − 1: s_n := remainder of (s_n−1)² − 2 upon division by M

• M is prime precisely when s_p−1 = 0

This is called the Lucas-Lehmer test

Edouard Lucas (1842–1891), Derrick H. Lehmer (1905–1991)

Other special primes: Fermat primes. Of the form 2^M + 1 For such a number to be prime, M must be a power of 2 Fermat thought that all numbers 2²ⁿ + 1 are primes

True for n = 0 upto n = 4. False for 5 ≤ n ≤ 32 2³² + 1 = 641 × 6700417

No example of a Fermat prime beyond 2¹⁶ + 1 known!

Testing primality is nowadays considered ‘easy’

(for numbers of upto thousands, not millions, of digits)

However, factoring composite numbers, is considered ‘hard’

(already for numbers of 200 digits)

Henri Cohen & Hendrik Lenstra (Leiden university)

‘CL’ in the APR-CL primality test (1984)

Theoretical result, announced in 2002, published in 2004:

Manindra Agrawal, Neeraj Kayal, Nitin Saxena Theorem. ‘PRIMES’ is in P

Meaning: there is a computer program that determines whether a given input number n is prime

Required computer memory for running it: a constant times the fifth power of the number of digits of n

Required computing time: at most a constant times the sixth power of the number of digits of n

the AKS algorithm

Agrawal, Kayal & Saxena

(Nitin Saxena now is a postdoc at CWI, Amsterdam)

Two applications of prime numbers 1. D-H key exchance protocol

named after Whitfield Diffie and Martin Hellman (1976)

(also invented by British Intelligence GCHQ employee Malcolm Williamson in 1974, but classified)

2. RSA cryptosystem

named after Ron Rivest, Adi Shamir and Leonard Adleman (1977) (also invented by GCHQ employee Clifford Cocks in 1973, but classified)

Diffie & Hellman

Rivest, Shamir & Adleman

D-H key exchange: two parties A and B have a ‘pincode’ (a large number) a and b, respectively

The pincode has to remain secret for everybody

Now they wish to share a common secret ‘key’ (also a large number)

D-H: the parties agree upon a number (say, 777) and a large prime p. Everybody can hear the choice of the number and of p A computes the remainder of 777^a upon division by p and gives this to B

B computes the remainder of 777^b upon division by p and gives this to A

Everybody hears what is given from one to the other

Now A raises what he received, to the ath power and computes the remainder upon division by p. Similarly for B, with the bth power. This gives them a common, secret key: it equals the remainder of 777^ab upon division by p

RSA: take large (secret) primes p and q Everybody is allowed to know N := p × q

Someone with a (known to you) code number e, can inform you of the secret message m (also a number, below N ), by sending you the remainder r of m^e upon division by N

Since you know p and q and e, you can compute a number d with the property that d × e has remainder 1 upon division by (p − 1) × (q − 1)

Having received r, you compute the remainder of r^d upon division by N . This equals the secret message m

Back to theory. Groningen, 1939

Theorem (Van der Corput) There exist infinitely many triples of primes p₁ < p₂ < p₃ with p₃ − p₂ = p₂ − p₁

So, such primes are in ‘arithmetic progression’: denoting the smaller of the three by p, they are p, p + a, p + a + a

Examples: 3, 3 + 2, 3 + 4 and 5, 5 + 6, 5 + 12 and 7, 7 + 12, 7 + 24 et cetera

Johannes Gualtherus van der Corput (1890–1975) Professor in Groningen between 1923 and 1945

Afterwards, first director of ‘Mathematisch Centrum’ in Amster- dam

In 2004, Van der Corput’s theorem was superseded by

Theorem (Green & Tao) There exist arbitrarily long sequences of primes in arithmetic progression

So, for any positive n, there is a prime p and a positive a such that all numbers

p, p + a, p + 2a, p + 3a, . . . , p + na are all prime

Terence Tao & Ben Green

One of the modern, open problems about primes: the ABC- conjecture

Read Nieuw Archief voor Wiskunde, March 2007

Currently, almost 4000 Dutch high school students provide com- puter time for the experiment “Reken mee met ABC”